ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A three shell after arc reduction

Example

A four-arc face has one exterior arc and three internal arcs. If its perimeter is $N$ and the internal lengths $a, b, c$ are each less than $N / 6$ , its exterior length exceeds $N / 2$ . Concretely use $N = 19$ and $a = b = c = 3$ , so the exterior arc has length $10$ .

This is a local shell length ledger. It does not assert a globally labelled $C^{'} (1 / 6)$ presentation realizing these lengths.

Facts & Assumptions

Given: The four-arc face, with perimeter $N > 0$ and positive internal lengths $a, b, c < N / 6$ .

[F1]

Arc reduction preserves word lengths and boundary incidence counts (Arc reduction and combinatorial curvature of a disc diagram).

[F2]

A three-shell consists of one exterior arc and three complementary internal arcs (Boundary spur or at most three shell from curvature).

Verification

1.1givenF1F2algebra

By [F1] and the three-shell description [F2], the exterior length is $N - a - b - c$ . The hypotheses give $a + b + c < 3 N / 6 = N / 2$ , hence $N - a - b - c > N / 2$ . Equality is excluded because all three piece bounds are strict.

2.1step 1.1algebra∎

For the displayed instance, $3 < 19 / 6$ because $18 < 19$ , and $19 - 3 - 3 - 3 = 10 > 19 / 2$ . Thus the interior part has length nine and the exposed part length ten, verifying the asserted instance.

Diagram

tikz \begin{tikzpicture} \draw[very thick] (0,0) -- node[above] {exterior: 10} (4,0); \draw (4,0) -- node[right] {3} (3,-2) -- node[below] {3} (1,-2) -- node[left] {3} (0,0); \end{tikzpicture}

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Touikan Definition 3.5.3 and Figure 3.5.4 (standard reference, not scraped)